VERTICAL ALIGNMENT

Spring 2015

Vertical Alignment

Geometric Elements of Vertical Curves

Vertical Grades

Passing Lanes

Sight Distance

Vertical Alignment

Highway engineers generally separate the characteristics of variations in typography according to the terrain: Level terrain: highway sight distances, as

governed by both horizontal and vertical restrictions, are usually long or can be made without construction difficulty.

Rolling terrain: natural slopes consistently rise above or fall below the road grade, and occasional steep slopes offer some restriction to normal alignment.

Mountainous terrain: longitudinal and transverse changes in the elevation of the ground are usually abrupt, and benching and side hill excavation are frequently needed.

Vehicle Operational Characteristics

Passenger cars: Grades as steep as 4% to 5% generally do not affect speed of most vehicles (may affect some compact/subcompact vehicles)

Trucks: Effects on speed much more important Maximum speed on upgrade is determined

length and steepness of the grade, and the truck’s weight/power ratio (gross weight/engine power)

Speed distance Curves for Trucks

Crawl Speed

Operational Characteristics of Trucks Travel time (and, therefore speed) of

trucks on grades is directly related to the weight/power ratio

Trucks with same weight/power ratios have similar operating characteristics

Units are kg/kW (metric) or lb/hp (U.S.) Trucks with a ratio above 200 lb/hp have

acceptable operating characteristics from the standpoint of highway users

Improve performance: lower weight and/or increase power

Control Grades for Design

Maximum Grades: 5% is considered adequate for design speed

of 70 mph 7% to 12% is considered acceptable for

design speed of 30 mph; if more important highways 7% or 8% should be used as the max

Values should fall between these extremes for other design speeds

Can use 1% steeper if the upgrade length is below 500ft

Use maximum design grade very infrequently

Control Grades for Design Minimum Grades:

The minimum grade is provided for drainage purposes

Typical 0.5% to 0.3% (for high-type pavement)

Particular attention should be given to the design of storm water inlets and their spacing

Climbing Lanes

Climbing lanes are increasingly used to decrease the amount of delay and improve safety (especially for 2-lane highways)

There are not designated as three-lane highways, but as a two-lane highways with an added lane

Climbing lanes are designed for each direction independently of each other

Where climbing lanes are provided, there has been a high degree of compliance by drivers

Climbing Lanes

Climbing Lanes

Criteria for two-lane highways: Upgrade traffic flow rate in excess of

200 vehicles per hour Upgrade truck flow rate in excess of 20

vehicles per hour A reduction in 10 mph or greater in

operational speed of trucks Level-of-service E or F (computed by

HCM) A reduction of two or more LOS between

entrance and upgrade segments

Climbing Lanes

Location of climbing lane Depends on the where the truck will reduce its speed by

10 mph Should extend the climbing lane beyond the crest

vertical curve to allow trucks to accelerate to previous speed

Make sure the climbing lane is wide enough Make use of signs “Slower Traffic Keep Right” or “Trucks

Use Right Lane”

Optimum length: 0.5 to 2.0 miles Taper length at the end: L=WS (W=width in ft,

S=speed in mph) Note: climbing on multilane highways usually not

justified (usually based on capacity analysis)

Vertical Curves

There are two types of vertical curves: Crest curves Sag curves

Design controlled by stopping sight distance

On occasion, decision sight distance may be needed

Vertical curves should result in a design that is safe, comfortable in operation, pleasing in appearance, and provide adequate drainage

Vertical Curves

Vertical Curves

Some design issues: The rate of change of grade (defined as the K-

Value: K = L /|A| ) should be kept within tolerable limits

Appearance can be important: short vertical curves may give the appearance of sudden break in profile

For sag curves: should retain a grade no less than 0.30% within 50 ft from the level point (i.e., changes from negative to positive grades)

The use of asymmetrical vertical curves may be required (e.g., sight restriction in sag curve)

Vertical Curve

y = 4E(x/L)2

K = L / |A|

A = G2 – G1

E = M = A L / 800

Rate of change of grade

External Distance

Offset

Ele. of P = [ele. Of VPC + (G1 / 100) x] + y

X = L|G1|/ (|G1 – G2|) (high or low point)

Vertical Point of Curvature

Vertical Point of Intersection

Vertical Point of Tangency

Vertical CurveA 600-ft vertical curve connects a +4% grade to a -2% at station 25+60.55 and elevation 648.64 ft. Calculate the location and elevation of the PVC, the middle of the curve, the VPT, and the curve elevation at stations 24+00 and 27+ 00.

Vertical CurvePartial answer:

A = -2 - (+4) = -6%

K = 600/|-6| = 100

E = -6 x 600 / 800 = -4.5 ft

Elevation of curve in the middle = 648.64 – 4.5 = 644.14

Using the curve elevation equation:

Ele P = [ele VPC + (G1/100) x] + y

644.14 ft = [ele VPC + 4/100 x 300] – 4.5

Ele VPC = 636.64

Station 22+60.55

Vertical Curve

Vertical Crest Curve

Vertical Crest Curve

Equation below is used for h1=3.5 ft and h2=2.0 ft

Vertical Crest Curve

For a design speed of 50 mph, determine the minimum length of a crest vertical curve with A=-4%. Assume h1 = 3.5 ft and h2 = 2.0 ft.

Vertical Crest Curve

For a design speed of 50 mph, determine the minimum length of a crest vertical curve with A=-4%. Assume h1 = 3.5 ft and h2 = 2.0 ft.

Stopping Sight Distance for 50 mph:

From Exhibit 3-1, SSD = 425 ft

S < L

L = 4 x 4252 / 2158 = 334 ft

S > L

L = 2 x 425 – 2158 / 4 = 310 ft

The length of curve is 310 ft since S > L

Vertical Crest Curve Design controls for crest vertical curves:

The minimum lengths of vertical curve for different values of A (G2 – G1) to provide the minimum stopping sight distance is provided in Exhibit 3-75

K = L /|A| (K= rate of vertical curvature, L=length of curve, A=difference in grade)

K is used to compare different curves; it covers all combinations of A and L for any one design speed

Minimum length: Lmin = 3V (V = mph, L = ft)

Vertical Crest Curve

Vertical Crest Curve

Previous example:

310 ft

Vertical Crest Curve

Vertical Crest Curve

Passing Sight Distance

Same principle as SSD, but with different values: h2 = 3.5 ft and S=PSD (Exhibit 3-7)

The lengths are 7 to 10 times higher

Vertical Sag Curve

The lengths are established based on four criteria: Headlight sight distance (h= 2 ft,

1o divergence angle) Passenger comfort (max acc = 1

ft/s2) Drainage control (minimum 0.3%) General appearance (same as

crest curve)

Vertical Sag Curve

Vertical Sag Curve

Vertical Sag Curve

Minimum Length for comfort:

Vertical Sag Curve

For a design speed of 60 mph, determine the length of the sag vertical curve with A = +10%

Vertical Sag Curve

For a design speed of 60 mph, determine the length of the sag vertical curve with A = +10%

Stopping Sight Distance: 570 ft

For S < L

L = 10 x 5702 / (400 + 3.5 x 570) = 1,357 ft

Check for comfort:

L = 10 x 602 / 46.5 = 774 ft (L = 1,357 ft is higher)

Vertical Sag Curve

Vertical Sag Curve

1,357 ft

Vertical Sag Curve

Vertical Sag Curve

Since many sag vertical curves are used at grade separated locations (overpass, bridges), highway engineers need to determine if the under passing structure creates a visibility impediment

Can use a graphical method or the equations provided in the text

Vertical Sag Curve

Vertical Sag Curve

S > L:

Vertical Sag Curve

S < L:

General Controls

Smooth gradeline with gradual changes is the preferable design alignment

“Roller-coaster” type of profile should be avoided (old roads)

A “broken-back” gradeline should be avoided

Sag vertical curves should be avoided in cuts unless adequate drainage can be provided

General Controls

Combinations of horizontal and vertical alignment: Horizontal and vertical alignments should not

be designed independently Highway cost is an important issue; thus, you

often need to design them together Vertical curvature superimposed on horizontal

curvature generally results in more pleasing facilities

Sharp horizontal curvature should not be introduced near the top of a pronounced crest vertical curve

See other design considerations in AASHTO Green Book